Triplet Josephson Effect with Magnetic Feedback in a Superconductor-Ferromagnet Heterostructure

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Triplet Josephson Effect with Magnetic Feedback

in a Superconductor-Ferromagnet Heterostructure

V. Braude and Ya. M. Blanter

Kavli Institute of Nanoscience, Delft University of Technology, 2628 CJ Delft, The Netherlands

(Received 18 June 2007; published 19 May 2008)

We study the ac Josephson effect in a superconductor-ferromagnet heterostructure with a variable magnetic configuration. The system supports triplet proximity correlations whose dynamics is coupled to the magnetic dynamics. This feedback dramatically modifies the behavior of the junction. The current-phase relation becomes double periodic at both very low and high Josephson frequencies !J. At

intermediate frequencies, the periodicity in !Jtmay be lost.

DOI:10.1103/PhysRevLett.100.207001 PACS numbers: 74.78.Fk, 72.25.Ba, 74.50.+r

Spin-dependent transport through hybrid structures combining ferromagnets (F) and normal metals has re-cently attracted a lot of interest, motivated by the prospect of potential technological applications in the field of spin-tronics [1]. Particular attention is given to two complemen-tary effects involving mutual influence between electric current and magnetic configuration. The first is giant mag-netoresistance [2] in which the conductance is much larger when different magnetic regions have their magnetic mo-ments aligned than when they are antialigned. The opposite effect is the appearance of torques acting on magnetic moments when an electric current flows through the system [3]. These nonequilibrium current-induced torques appear due to nonconservation of spin currents accompanying a flow of charge through ferromagnetic regions. They allow manipulation of the magnetic configuration, including switching between the opposite directions or steady-state precession, without application of magnetic fields [4]. The two effects combined promise important practical applica-tions in nonvolatile memory, programmable logic, and microwave oscillators.

A conceptually different situation occurs when a ferro-magnet is coupled to a superconductor (S), since the spin current through the superconducting part vanishes [5]. This additional constraint modifies the nonequilibrium torques, opening the possibility of perpendicular alignment of mag-netic moments. Furthermore, if a magmag-netic structure is contacted by two superconductors, the proximity effect may be present, leading to a finite Josephson current through the structure at equilibrium. The torques generated by this current correspond to an equilibrium effective exchange interaction between the magnetic moments which can be controlled by the phase difference between the superconductors [6]. The same mechanism enables the reciprocal effect in which the supercurrent depends on the magnetic configuration.

For a uniform ferromagnet, the observation of these effects requires a very thin magnetic layer, since the prox-imity effect is suppressed at short distances. However, in nonuniform ferromagnets, a long-range proximity effect

can exist due to triplet superconducting correlations [7]. This triplet proximity effect (TPE) strongly enhances the associated Josephson current. It is important that TPE essentially depends on the magnetic configuration of the system [7]. Hence S/F multilayers exhibiting TPE are especially suitable for studying the Josephson-induced magnetic exchange interaction. By varying the relative magnetization directions of different magnetic regions, one can control the supercurrent flowing through the struc-ture. Then, if the magnetic configuration is allowed to respond to the Josephson-current induced torques, it cre-ates feedback for the supercurrent and considerably modi-fies it.

In this Letter, we consider the magnetic exchange inter-action induced by Josephson currents in a dirty S/F hetero-structure exhibiting TPE. We show that this interaction favors noncollinear magnetic configurations, and the pre-ferred direction depends continuously on the supercon-ducting phase difference . Thus, the static magnetic configuration can be controlled by the applied phase dif-ference. We then consider the influence of feedback from the magnetic moments on the ac Josephson effect. We demonstrate that the magnetic system exhibits a range of different behaviors, from simple harmonic oscillations to fractional-frequency periodic behavior and chaotic motion. The magnetic feedback complicates the behavior of the current in the time domain, making it generally impossible to express it in terms of a current-phase relation. This is in contrast with the ac Josephson effect without the magnetic feedback, where the time dependence of the supercurrent J is determined by the current-phase relation (J J_csin for TPE in diffusive systems, with the exception of the magnetic configuration with mutually perpendicular direc-tions where a transition between ‘‘0’’ and ‘‘’’ states occurs [8]).

On the other hand, we find that both in the low- and high-frequency limit the current-phase relation becomes mean-ingful, with the current exhibiting a double-phase depen-dence, J / sin2t [9] or J / cos2t. The critical current in the low-frequency regime is of the order of the PRL 100, 207001 (2008) P H Y S I C A L R E V I E W L E T T E R S 23 MAY 2008week ending

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value E_Th=eR_n, characteristic for diffusive systems, E_Th and R_nbeing Thouless energy and normal-state resistance of the junction, respectively. The unusual cosine depen-dence of the Josephson current appears when Gilbert damping is important in the magnetic dynamics, breaking the time-reversal symmetry. At high frequencies !_J, the magnetization cannot effectively follow the phase varia-tion, leading to a 1=!2

J suppression of the effective

Josephson coupling. At even higher frequencies, the damp-ing is dominant, and the frequency dependence becomes 1=!_J. The presence of damping is expressed in the appear-ance of a dc component of the current leading to a finite resistance.

The system. —We consider the minimum discrete setup

exhibiting TPE (Fig.1). Two magnetic regions, 1 and 3, are adjacent to the superconducting reservoirs that induce proximity mini-gaps _1;3in them. Between these regions there is an additional single-domain magnetic region, 2, whose length is much larger than the coherence length h

(and shorter than the coherence length in the absence of exchange field _N), where triplet superconducting correla-tions are induced. This region is assumed to be weakly polarized (metallic), so that both spin directions are present at the Fermi surface. The magnetic regions are character-ized by the exchange energies h_i, while the magnetization directions n_i are specified by the angles ₁, ₃, and  (Fig.1). Assuming that the conductances of these regions are much higher than those gn

1;3of the connectors between them, our system can be described by a circuit-theory model for TPE [8].

Let us now discuss the minimum requirements for the magnetic feedback. If all three directions of the magneti-zation n₁_–₃ can freely rotate, they always prefer to be aligned, thus suppressing the TPE. The same situation occurs if any two of these vectors are free. In order to induce TPE, we thus need to fix the directions of two of the vectors. If only the middle vector n₂is allowed to rotate, it will assume its equilibrium position along one of the two bisectors between the fixed directions n₁ and n₃. On the

other hand, if the magnetic vector of one of the outer regions is allowed to rotate, an interesting situation arises when its equilibrium direction continuously depends on the superconducting phase difference across the junction. In order to explore this situation, we assume that n₁ and n₂ are fixed, e.g., by pinning to an antiferromagnetic sub-strate, or by geometrical shaping, with the angle between them being 1. Magnetization n3 is free to rotate, with region 3 separated by a normal spacer from region 2 in order to avoid exchange coupling between them.

In accordance with the model assumptions, regions 1 and 3 act as effective S/F reservoirs; hence, their energies are independent of the magnetic configuration. On the other hand, triplet superconducting correlations extending through region 2 are very sensitive to the magnetization directions. The configuration-dependent part of the energy can be found by integrating over the density of states (DOS) in region 2. The DOS for each spin direction is given by [8] ";#" 0 2 Re 1 a 2 1 a23 2a1a3cos b₁ b₃ i=E_Th2 1=2 ; (1)

where 0 is the normal-state DOS, ak gn_kjkj sink=gn1 gn3 h2_k jkj2 q , and bk gnkhk= gn 1  gn3 h2_k jkj2 q

. The energy is given by a logarith-mic integral, and the main contribution comes from 

E_Th. In the leading order one obtains

E 0v2 2 log cut ETh E2_Tha2 1 a23 2a1a3cos cos; (2) where v₂is the volume of the magnetic region 2 and _cut’ mini; hi i is a cutoff energy. This expression can be

written in a form presenting explicitly the dependence on the orientation angles 3 and ,

E p2

3sin23 2p1p3sin3cos cos; (3) with p_1;3being effective exchange couplings for the mag-netic vector n3. The stable configuration is achieved when all magnetization directions are in the same (xz) plane, and n3 is tilted with respect to n2,

sin₃ p₁=p₃j cosj: (4) This angle depends continuously on the applied supercon-ducting phase difference , while the angle assumes the values 0 or so that the product cos cos is negative. In fact, there are two stable directions, given by the angles ₃ and ₃. In what follows we treat them as equivalent, since they correspond to the same current. The energy of the stable configuration is given by E_min p2

1cos2. Hence, allowing the magnetization direction n3 to orient itself along the stable direction leads to the current-phase relation J J_csin2.

S

n

₁

n

₂

n

3

z

x

y

n

₁

θ

1

θ

n

3 3

χ

n

2

FIG. 1. The experimental setup.

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Low frequencies. —When a small voltage V is applied to

the structure, such that the corresponding Josephson fre-quency !_J 2eV=@ is much smaller than the character-istic frequency of the magnetic system !_m(see below), the vector n₃ follows the stable direction given by Eq. (4), performing slow oscillations in the x-z plane. The alter-nating Josephson current oscillates with the double fre-quency

J 2e

@ p21sin 4eV

@ t; (5)

while the critical current remains of the same order of magnitude as in the case with a fixed magnetic configuration.

For higher Josephson frequencies, the variation of n₃is no more limited to the x-z plane. Instead, the magnetiza-tion performs a variety of nonharmonic momagnetiza-tions whose frequency may be a multiple or a fraction of the driving frequency !_J [Figs. 2(a), 2(c), and 2(d)]. For certain trajectories the time average of ₃ is finite [Fig. 2(c)], corresponding to a tilt of n₃ away from the equilibrium in response to an applied voltage. Within some frequency intervals, the motion is chaotic, as shown in Fig.2(b). In these intermediate regimes, the Josephson current shows a complicated time dependence which is generally not peri-odic in 2=!J. Hence this dependence cannot be

parame-trized in terms of the phase. Instead, one can speak of a Josephson current with a time-dependent coupling.

High frequencies. —If the Josephson frequency is much

higher than the magnetic frequencies, the vector n₃cannot effectively follow the fast oscillations of the potential, and the time-averaged potential seen by n₃has a minimum for

n₃ k z. The motion of n3 can be determined by expanding n₃ z n and using a linearized Landau-Lifshits-Gilbert (LLG) equation,

 _n z Heff n; (6)

where is the gyromagnetic ratio, is the damping coefficient, H_eff @E=@m₃ is the effective field, and m₃ is the magnetization density of region 3.

When Gilbert damping is negligible, the trajectory of n₃ has a very low aspect ratio, so that the motion is almost completely confined to the y axis. It is given by

nx 2_p 1p33@2 e2V2m2₃ cos 2eVt @ ; ny p1p3@ eVm₃ sin 2eVt @ : Thus at high frequencies, n3 precesses in phase with the voltage pumping. This leads to an increase in the Josephson energy, and, correspondingly, a negative Josephson current, J 2@ e _p 1p23 Vm3 2 sin4eVt @ : (7)

Hence in the high-frequency regime the system shows not only frequency doubling, but also an effective junction behavior. The magnitude of the current is suppressed as V2_{as shown in Fig.}₃_.

The neglect of damping is justified as long as !_J

!_m p2

3=m3. When the voltage is high enough, this condition is not satisfied anymore, and the dissipation starts to be important. In the opposite case !_J  !_m, the motion of n₃ is determined by the driving against the damping force, −1 −0.5 0 0.5 1 −0.2 −0.1 0 0.1 0.2 0.3 n x ny ωJ=0.1 T= T₀ a) −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 n x ω_J=0.3 b) −0.8 −0.6 −0.4 −0.2 −1 −0.5 0 0.5 1 n x ny ωJ=0.8 T=2T₀ c) −1 −0.5 0 0.5 1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 n x ω_J=1.3; T=3T₀ d)

FIG. 2 (color online). Trajectories of the magnetization vector in the x-y plane for different Josephson frequencies (given in units of !m). Trajectory (b) is chaotic, while trajectory (c) has a

finite zero-frequency component for nx. Here T is the period of

the trajectory, and T0 2=!J. For comparison, the

low-frequency trajectory lies entirely on the x axis.

100 101 10−4 10−3 10−2 10−1 ω_J/ω m amplitude ∼ ω_J−1_↑ ← ∼ ω_J−2

FIG. 3 (color online). The absolute value of the Josephson-current harmonics proportional to sin2!Jt(asterisks) and to 1

cos2!Jt (dots). Solid lines are fits 1=!J and !2J . The data

points are obtained from numerical integration of the full (non-linear) LLG equation.

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n p1p3@ eVm3

^x ^y sin2eVt

@ : (8)

Then the Josephson current is given by

J 2 p 2 1p23 m3V 1 cos4eVt @ : (9)

Note the unusual cosine dependence on the phase. It occurs since the time-reversal symmetry is broken by the dissipa-tion in this regime. Due to the same reason, a zero-frequency component of the current appears, signifying the onset of a finite nonlinear resistance across the struc-ture. Since this regime is governed by the damping, the current amplitude is proportional to , while the suppres-sion 1=V is weaker in this regime (Fig.3).

To estimate the magnetic dynamics frequency !m, we

use typical values ETh 1 meV, 0 1=eV=atom, and m3 1B=atom, where B is the Bohr magneton. Then !m v2=v3 GHz, where v2;3are the volumes of the cor-responding magnetic regions. As this frequency is quite low, observation of the high-frequency regimes should present no difficulty. On the other hand, the low-frequency ac regime would require extremely low voltages, below 1 V. A reasonable alternative would be incorporating the structure in a superconducting loop and measuring the Josephson current as a function of the applied flux.

Applicability of our model requires that any magnetic anisotropy of part 3 should be smaller than the proximity-induced energy, Eq. (2). With the above values of the parameters it is of the order of 104_v

2 J=m3; thus, one should choose materials with a low value of the crystalline anisotropy, such as permalloy. Generally, the observation of the effect would be easier in materials with the low exchange field. Finally, we emphasize that the properties discussed above are specific for metallic systems. In half-metals, the behavior will be very different. Thus, in the low-frequency regime n₃precesses around n₂at a constant angle ₃, while the Josephson current vanishes.

Conclusions.—We have considered the ac Josephson

effect in a SFS structure with magnetic dynamics coupled to the dynamics of superconducting correlations. The mag-netic configuration in the structure was assumed to be nonuniform so that the structure exhibits TPE. Variation of the magnetic configuration is shown to essentially mod-ify the current behavior that can be observed in the appear-ance of fractional Shapiro steps. Thus measurement of the Josephson current would provide information about the coupling and self-consistent feedback dynamics between the superconducting and magnetic degrees of freedom. The coupling also allows control of the magnetization direction by means of applied voltage or superconducting phase. In the low-frequency limit, the magnetization follows the immediate potential minimum, leading to a sin2 current-phase relation. The critical current has the same order of magnitude ETh=eRn as the one due to the usual

singlet proximity effect in dirty structures. In the high-frequency regime, as long as the damping is not important, the Josephson current is negative, corresponding to a

-junction behavior. It is suppressed by a factor !_m=!_J2 _{relative to the low-frequency regime. At even} higher frequencies, Gilbert damping starts playing the major role in the dynamics. Then the time-reversal sym-metry is broken and the current-phase relation takes an unusual cosine form. In addition, a dc component of the current appears, manifesting itself in a finite resistance. The current suppression becomes weaker in this regime. The even current-phase relation has been recently pre-dicted in Ref. [13] for a system with spin-active interfaces. Note, however, that in our case for high frequencies the current-phase relation is not meaningful, and the even time dependence appears due to the damping.

The authors thank G. E. W. Bauer and Yu. V. Nazarov for useful discussions. This work was supported by EC Grant No. NMP2-CT2003-505587 (SFINX).

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